Polynomial in $\mathbb{C}^2$ Let $p(z,w) = a_0(z) + a_1(z) w + \dots + a_k(z) w^k$, where $k \ge 1$ and $a_i (z)$ are non-constant polynomial of $z$. Which of the following shall be correct for the set $$A = \{(z,w) \in \mathbb{C} \times \mathbb{C} : p(z,w) = 0\} ?$$
A. Bounded with empty interior.
B. Unbounded with empty interior.
C. Bounded with nonempty interior.
D. Unbounded with nonempty interior.
My Attempt
I can conclude it if we consider $\mathbb{R}$ instead of $\mathbb{C}$. $p(x,y) = 0$ when each $a_i(x) = 0$ i.e. $x$ is a common zero of $a_i$. If $x = m$ be such a zero then the set $\{(m,y) : y \in \mathbb{R}\}$ is unbounded in $\mathbb{R}^2$ with empty interior. $A$ will be union of such disjoint sets. So $A$ should be unbounded with empty interior.
The situation should be different in $\mathbb{C}$. But I do not know analysis of sevaral complex variable. What will be in $\mathbb{C}$? Is my approach for $\mathbb{R}^2$ is correct?
Thank you for your help.
 A: You don't need any complex analysis in several variables for this, the fundamental theorem of algebra is enough.
For any fixed $z \in \mathbb{C}$ with $a_k(z) \neq 0$, there can be only finitely many $w\in \mathbb{C}$ with $p(z,w) = 0$, since $p(z,\,\cdot\,)$ is a polynomial of degree $k$. Thus none of these points can be an interior point. But $a_k$ has only finitely many zeros, so none of the points $(z,w)$ with $a_k(z) = 0$ can be an interior point either.
Thus $A$ has empty interior. (With a little theory of several complex variables, this part would reduce to an invocation of the identity theorem: if $A$ had nonempty interior, $p \equiv 0$.)
Further, for all $z$ with large enough modulus, $a_k(z) \neq 0$, so $p(z,\,\cdot\,)$ has precisely $k$ zeros in $\mathbb{C}$ (counting multiplicity), in particular, $A \cap \{z\}\times \mathbb{C}\neq \varnothing$, so $A$ is unbounded.
A: $z^{k+1}+z^kw+...+z^2w^{k-1}+zw^k$ has unbounded zero set. So $1$ and $3$ are not true .
Also, if the interior were not empty, there would've existed $U\times V$ in $\mathbb C^2$ where $p(z,w)$ is identically zero. Fix $z$ in $U$ such that it is not a root of $a_k$. You get a non constant polynomial in one variable $w$ vanishing on $V$, a contradiction. So the interior must be empty.
